Bubble-and-grain method and criteria for optimal positioning inhomogeneities in topological optimization

In the article we propose the enhancement of the topology optimization method, which uses an iterative positioning, orientation and hierarchical shape optimization of subsequently introduced elastic inhomogeneities. The inserted elastic inhomogeneities could be more or less compliant than the elastic medium of the structural element being optimized. One extreme case of the inhomogeneity is the cavity of zero stiffness (bubble), while the other limit corresponds to the absolutely rigid inhomogeneity (grain). This extension of the topology method requires the generalization of topological derivatives. The topological derivative is an instrument for solving topology optimization problems. Namely, the topological derivative quantifies the sensitivity of a problem when the domain under consideration is perturbed by changing its topological genus. In this article we represent the generalized topological derivatives exploiting the Eshelby approach of effective inhomogeneity. For this purpose we study sensitivity of the optimization functional to the placement of infinitesimally small elliptical inhomogeneity. The sensitivity to the infinitesimal translation of inclusion is quantified by the characteristic function. The infinitesimally small inhomogeneity must be inserted at the point, where the characteristic function attains its extreme value. Next, we examine the sensitivity of the Lagrangian to orientation of ellipse and determine its optimal orientation. Finally, we express the optimal eccentricity of ellipse as the function of averaged principal strains in inhomogeneous medium. The compliance functional plays the role of optimization criterion. Using adjoint variables technique of variational calculus, the results could be extended for arbitrary integral functionals.